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May 10th, 2012, 02:43 PM  #1 
Member Joined: Mar 2012 Posts: 60 Thanks: 0  The shape of a parabola
I've been trying to wrap my head around why all quadratic equation parabolas with a leading coefficient of 1 (or just have the same leading coefficient) have the exact same shape, just a different location around the Cartesian graph. ex X^2 , X^22x, and X^210x All the parabolas will have the exact same shape, and increase the exact same amount from their respective vertexs 0,0 1,1 1,1 2, 0 increase of 1 2,4 3,3 of 3 3,9 4,8 of 5 It seems almost counterintuitive that the graphs will always have the exact same shape, and move away from the vertex's at the exact same pace. I mean, there must be some pattern that when you are at the point where you increase or decrease X by any amount, Y will go up (plain English the vertex) then it will precisely follow the parent graph X^2. I would like to visualize or grasp some concept of why this is. My squared drawings of dots were less than helpful. 
May 10th, 2012, 03:00 PM  #2 
Member Joined: Mar 2012 Posts: 60 Thanks: 0  Re: The shape of a parabola
Complete the squares and everything will be perfectly clear 
May 10th, 2012, 03:15 PM  #3  
Member Joined: Mar 2012 Posts: 60 Thanks: 0  Re: The shape of a parabola Quote:
x^22x (x1)^2= 1 (x1)^21=x^22x vs x^2 Okay so the graph will be the exact same, except in this case x will have to be 1 more to the right to compensate for the 1, and every single value of Y will be 1 less because of the constant. So now it is quite obvious the graph will behave in the exact same way. Thanks for my noob reminder!  
December 24th, 2012, 09:22 PM  #4  
Global Moderator Joined: Dec 2006 Posts: 20,941 Thanks: 2210  Quote:
 
December 28th, 2012, 06:22 PM  #5  
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  Re: Quote:
 
December 30th, 2012, 08:35 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,941 Thanks: 2210 
Usually, yes, but the original poster seemed to have congruence in mind.

December 30th, 2012, 09:59 PM  #7  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: The shape of a parabola Hello, maxgeo! Did you understand fredde_fisk's suggestion? Quote:
Complete the square: [color=beige]. . . . . . . . [/color] [color=beige]. . . . . . . .[/color] [color=beige]. . . . . . . .[/color] [color=beige]. . . . . . . .[/color] [color=beige]. . . . . [/color]  

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